Surface Area

Surface Area
A skyline is a view of the outline of
buildings or mountains shown on
the horizon. You can see skylines
during the day or at night, all over
the world. Many cities have
beautiful skylines. City planners
have to consider much more than
just how the skyline will look when
they design a city.
In the skyline shown in the picture,
what shapes do you see? What
three-dimensional objects can
you identify?
In this chapter, you will learn how to
draw and build three-dimensional
objects and how to calculate their
surface areas.
What You Will Learn
φ
φ
φ
to label and draw views of 3-D objects
φ
to solve problems using surface area
2
to draw and build nets for 3-D objects
to calculate the surface area for prisms
and cylinders
MHR • Chapter 5
Key Words
• face
• edge
• vertex
• rectangular prism
• net
Literacy
• triangular prism
• right prism
• surface area
• cylinder
Link
You can use a Verbal Visual Chart (VVC) to help
you learn and remember new terms.
Copy the blank VVC into your math journal or
notebook and use it for the term, rectangular
prism.
• Write the term in the first box.
• Draw a diagram in the second box.
• Define the term in the third box. The glossary
on pages XXX–XXX may help you.
• In the fourth box, explain how you will
remember the term and what it means.
Consider using an example, a characteristic, a
memory device, or a visual.
Term
Diagram
Definition
How I Will
Remember It
Chapter 5 • MHR
3
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Making The Foldable
•
•
•
•
•
•
Step 4
Make the paper from Step 3 into eight booklets
of 4 pages each.
11 × 17 sheet of paper
ruler
glue or tape
four sheets of blank paper
scissors
stapler
Step 5
Collapse the Foldable. Title the faces of your
Foldable. Then, staple the booklets onto each
face, as shown, and add the labels shown.
Step 1
Fold over one of the short sides of an 11 × 17 sheet
of paper to make a 2.5 cm tab. Fold the remaining
portion of paper into quaters.
5.1
V
Dim iews
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ehre ts
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5.2 ensio
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No
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at
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Wo
rd
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ork
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at
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Wo
to
On
IN
p!
tU
pI
as
Ide
a
Wr
s
Key
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2.5 cm
Step 2
Use glue or tape to put the paper
together as shown in the diagram.
If you use glue, allow it to dry
completely.
Step 3
Fold each of four sheets of blank paper into
eighths. Trim the edges as shown so that each
individual piece is 9.5 cm × 6 cm. Cut off all
folded edges.
9.5 cm
As you work through each section of Chapter 5,
take notes on the appropriate face of your
Foldable. Include information about the examples
and Key Ideas in the Notes section. If you need
more room, add sheets of paper to your booklet.
List and define the key words in the Key Words
booklet. Use visuals to help you remember the
terms.
Keep track of what you need to work on. Check off
each item as you deal with it.
6 cm
4
Using the Foldable
MHR • Chapter 5
As you think of ideas for the Wrap It Up!, record
them on that section of each face of your Foldable.
MATH LINK
City Planning
When city planners design communities, they consider the
purpose of the buildings, the width of the streets, the
placement of street signs, the design and placement of
lampposts, and many other items found in a city.
Communication and cooperation are keys to being successful,
because city planners have to coordinate and work with many
other people.
Imagine that you are a city planner for a miniature community.
Discuss your answers to #1 and #2 with a partner, then share
with your class.
1.
a) What buildings are essential to a new community?
b) What different shapes are the faces of these buildings?
2.
What other items are important to include in a community?
3.
Using grid paper, sketch all or part of an aerial view of a
community including the essential buildings your class
discussed. Make sure to include roads and any other
features that are important.
In this chapter, you
will work in groups to
create and design a
miniature community.
Math Link • MHR
5
Views of Three-Dimensional
Objects
Focus on…
<Ill 5-1: illustration of two
students split view; one girl making
an object with unit blocks, one boy
making a DIFFERENT object with the
same number of unit blocks. Bubbles
with dialogue, girl (Sable): “I used
ten blocks” (or how ever many is in
the picture), boy (Josh): “Sable,
what does yours look like if you look
at it from the top?”>
After this lesson,
you will be able to...
φ draw and label
φ
top, front, and
side views of
3-D objects
build 3-D
objects when
given top, front,
and side views
Sable and Josh are trying to build exactly the same three-dimensional
• 20 unit blocks
• masking tape
• isometric dot paper
A^iZgVXn
A^c`
To describe a
three-dimensional
(3-D) object, count its
faces, edges, and
vertices.
Face:
flat or
curved
surface
Edge: line
segment
where two
faces meet
Vertex: point
where three or
more edges meet
6
MHR • Chapter 5
(3-D) object. They each have the same number of blocks, but they
cannot see each other’s object.
Using a common vocabulary can help Sable and Josh build the
same object.
How can you describe and build three-dimensional objects?
1.
Work with a partner. Create a 3-D object using ten unit blocks.
Make sure your partner cannot see your object.
2.
Describe your completed object to your partner, and have your
partner try to build the same object. What key words did you
use that were helpful?
3.
Decide which faces will be the front and top of your object. Then
determine which faces are the bottom, left side, right side, and back.
You may wish to label the faces with tape. Then, describe your
object to your partner again. Was it easier to describe this time?
4.
Using isometric dot paper, draw what your object looks like.
Reflect on Your Findings
Do you need to know all the views to construct an object?
If not, which ones would you use and why?
b) Explain why you might need to have only one side view,
if the top and front views are also given.
c) Are any other views unnecessary? Are they needed to
construct the same object?
5. a)
Using isometric dot paper makes
it easier to draw 3-D shapes.
Follow the steps to draw a
rectangular solid.
1
3
2
4
Each view shows two
dimensions. When
combined, these views
create a 3-D diagram.
Example 1: Draw and Label Top, Front, and Side Views
Using blank paper, draw the top, front, and side views of these items.
Label each view.
a)
Tissue box
b)
Compact disk case
Solution
a)
top
b)
top
front
front
side
(end of the box)
side
5.1 Views of Three-Dimensional Objects • MHR
7
Using blank paper, draw the top, front, and
side views of this object.
Example 2: Sketch a Three-Dimensional Object When Given Views
Architects use top
views to draw
blueprints for
buildings.
These views were drawn for an object made of ten blocks.
Sketch what the object looks like.
top
side
front
Solution
Use isometric dot paper
to sketch the object.
An object is created using eight
blocks. It has the following top,
front, and side views. Sketch what
the object looks like on isometric
dot paper.
8
MHR • Chapter 5
top
front
side
Example 3: Predict and Draw the Top, Front, and Side Views
After a Rotation
The diagrams show the top, front, and side views of the computer tower.
top
front
side
You want to rotate the computer tower 90º clockwise on its base to fit
into your new desk. Predict which view you believe will become the
front view after the rotation. Then, draw the top, front, and side views
after rotating the tower.
This diagram shows
a 90° clockwise
rotation.
90°
Solution
The original side view will become the new front view after the rotation.
top
front
side
You can use a Draw
program to create
3-D objects.
Stand your MathLinks 8 textbook on your desk. Predict what the top,
front, and side views will look like if you rotate it 90º clockwise about
its spine. Then, draw the top, front, and side views after rotating
the textbook.
5.1 Views of Three-Dimensional Objects • MHR
9
• A minimum of three views are needed to describe a 3-D object.
• Using the top, front, and side views, you can build or draw a
3-D object.
top
side
front
1.
Raina insists that you need to tell her all six views so she can draw
your object. Is she correct? Explain why or why not.
2.
Are these views correct? Justify your answer.
front
top
side
c)
For help with #3 and #4, refer to Example 1 on
page XXX.
3.
Sketch and label the top, front, and
side views.
a)
b)
4.
Photo
Album
Choose the correct top, front, and side
view for this object and label each one.
A
10
MHR • Chapter 5
E
B
F
C
G
D
For help with #5, refer to Example 2 on page xxx.
5.
Draw each 3-D object using the
views below.
a)
b)
top
top
front
front
8.
Choose two 3-D objects from your
classroom. Sketch the top, front, and
side views for each one.
9.
Sketch the front, top, and right side
views for these solids.
side
side
a)
For help with #6 and #7, refer to Example 3 on
page xxx.
6.
front
front
front
c)
A television set has the following views.
top
b)
side
front
If you turn the television 90°
counterclockwise, how would the three
views change? Sketch and label each
new view.
7.
Choose which object
has a front view like this
after a rotation of
90º clockwise onto its side.
a)
b)
set of books
CD rack
10.
Describe two objects that meet this
requirement: When you rotate one object
90 degrees, the top, front, and side views are
the same as the top, front, and side views of
the other object that was not rotated.
11.
An injured bumblebee sits at a vertex of a
cube with edge length 1 m. The bee moves
along the edges of the cube and comes
back to the original vertex without visiting
any other vertex twice.
a) Draw diagrams to show the
bumblebee’s trip around the cube.
b) What is the length, in metres, of the
longest trip?
MATH LINK
Choose one of the essential buildings that you discussed for your new community
on page XXX. Draw and label a front, side, and top view.
5.1 Views of Three-Dimensional Objects • MHR
11
Nets of Three-Dimensional
Objects
Focus on…
After this lesson, you
will be able to...
φ determine the
φ
φ
correct nets for
3-D objects
build 3-D objects
from nets
draw nets for
3-D objects
rectangular prism
• a prism whose bases
are congruent rectangles
Shipping containers help distribute materials all over the world. Items
can be shipped by boat, train, or transport truck to any destination
using these containers. Shipping containers are right rectangular prisms .
Why do you think this shape is used?
•
•
•
•
grid paper
scissors
clear tape
rectangular prisms
(blocks of wood,
cardboard boxes,
unit blocks)
How do you know if a net can build a right rectangular prism?
Here are a variety of possible nets
for a right rectangular prism.
rectangular prism
net
• a two-dimensional
shape that, when
folded, encloses a
3-D object
net
12
cube
MHR • Chapter 5
Literacy
Link
A right prism has
sides that are
perpendicular to the
bases of the prism.
1.
Draw each net on grid paper.
2.
Predict which nets will form a right rectangular prism.
3.
Cut each net out along the outside edges and fold along the inside
edges, taping the cut edges to try to form a right rectangular prism.
4.
Do all the nets create right rectangular prisms?
5.
Place a right rectangular prism (such as a small cardboard box) on
a piece of blank paper. “Roll” the prism onto its faces, trace each
face, and try to draw another correct net. Your net should be
different from the examples you have already made.
Reflect on Your Findings
Compare the net you drew with those of three of your
classmates. What is the same and different about your nets?
b) Is there more than one way to draw a net for a 3-D object?
Explain your answer.
6. a)
Example 1: Draw a Net for a Three-Dimensional Object
A company asks you to create an umbrella stand
for large beach umbrellas. Draw the net for the
umbrella stand.
Strategies
Solution
Visualize what the umbrella stand would look like if you could cut it
open and flatten it. The net has one circle and a rectangle. When the
rectangle is curved around the circle, the net will form a cylinder with
an open top. The width of the rectangle is equal to the circumference
of the circle.
Change Your Point
of View
Refer to page xxx.
Draw a net for an unopened soup can.
5.2 Nets of Three-Dimensional Objects • MHR
13
Example 2: Build a Three-Dimensional Object From a Given Net
Before going to leadership camp, your group needs to put a tent
together. Can this net be folded to form the shape of a tent?
Strategies
Model It
Refer to page xxx.
triangular prism
Solution
Trace the net onto paper. Cut along the outside edges and fold
along the inside edges. Tape the cut edges together to try to build
a right triangular prism .
• a prism with two
triangular bases each
the same size and
shape
The net can be folded to form the shape of a tent.
Build a 3-D object using this net. What object
does it make?
14
MHR • Chapter 5
• A net is a two-dimensional shape that, when folded,
encloses a three-dimensional object.
net
cube
• The same 3-D object can be created by folding
different nets.
• You can draw a net for an object by visualizing what it would
look like if you cut along the edges and flattened it out.
1.
Both of these nets have six faces, like a cube. Will both nets
form a cube? Justify your answer.
net A
2.
net B
Patricia is playing the lead role in the school musical this year. She
missed Math class while she was performing. She cannot figure out
if a net will build the correct 3-D object, and asks you for help after
school. Show how you would help her figure out this problem.
For help with #3 to #5, refer to Example 1 on page xxx.
3.
Sketch a net for each object.
a)
b)
hockey puck
c)
chocolate bar
jewellery box
5.2 Nets of Three-Dimensional Objects • MHR
15
4.
Draw the net for each object. Label the
measurements on the net.
a)
7.
Match each solid with its net. Copy the
nets, then try to create the 3-D objects.
d = 30 mm
rectangular prism
A ream describes a
quantity of
approximately 500
sheets of paper.
78 mm
cylinder
b)
triangular prism
A
28 cm
Paper
500 Sheets
B
5 cm
21.5 cm
5.
Draw a net on grid paper for a rectangular
prism with the following measurements:
length is six units, width is four units, and
height is two units.
C
For help with #6 and #7, refer to Example 2 on
page xxx.
6. a)
D
Draw the net on grid paper, as shown.
Cut along the outside edges of the net
and fold to form a 3-D object.
E
b)
16
8.
A box of pens measures 15.5 cm by 7 cm
by 2.5 cm. Draw a net for the box on a
piece of centimetre grid paper. Then, cut
it out and fold it to form the box.
9.
You are designing a new mailbox. Draw
a net of your creation. Include all
measurements.
What is this object called?
MHR • Chapter 5
10.
Angela designed two nets.
12.
What colour is on the opposite side of each
of these faces?
a) purple
b) blue
c) red
Enlarge both nets on grid paper, and
build the 3-D objects they form.
b) What object does each net form?
a)
13.
11.
The six sides of a cube are each a different
colour. Four of the views are shown below.
How many possible nets can create a cube?
Sketch all of them. The first one is done
for you.
Hannah and Dakota design a spelling
board game. They use letter tiles to create
words. Tiles may be stacked (limit of four)
on top of letters already used for a word
on the board to form a new word.
a) Draw a 3-D picture of what these
stacked tiles might look like.
b) Draw a top view that illustrates the
stacked tiles for people reading the
instructions.
MATH LINK
When buildings are designed, it is
important to consider engineering
principles, maximum and
minimum height requirements,
and budget.
a) Create a 3-D sketch of two
buildings for your miniature
community, one that is a prism
and one that is a cylinder.
b) Draw a net of each building,
including all possible
measurements needed to
build your miniature.
5.2 Nets of Three-Dimensional Objects • MHR
17
Surface Area of a Prism
Focus on…
After this lesson,
you will be able to...
φ link area to
φ
surface area
find the surface
area of a right
prism
Most products come in some sort of packaging. You can help
conserve energy and natural resources by purchasing products that
• are made using recycled material
• use recycled material for packaging
• do not use any packaging
What other ways could you reduce packaging?
How much cardboard is needed to make a package?
• empty cardboard box
(cereal box, granola
box, snack box, etc.)
• scissors
• ruler
• scrap paper
Literacy
1.
Choose an empty cardboard box.
a) How many faces does your box have?
b) Identify the shape of each face.
2.
Cut along the edges of the box and unfold it to form a net.
Link
Exclude
overlapping
flaps.
The dimensions of an
object are measures
such as length,
width, and height.
18
MHR • Chapter 5
3.
What are the dimensions and area of each face?
4.
How can you find the amount of material used to make your
cardboard box?
Reflect on Your Findings
How did you use the area of each face to calculate the total
amount of material used to make your cardboard box?
b) Can you think of a shorter way to find the total area without
having to find the area of each face? Explain your method.
5. a)
Example 1: Calculate the Surface Area of a Right Rectangular Prism
Draw the net of this right rectangular prism.
b) What is the area of each face?
c) What is the surface area of the prism?
a)
6 cm
surface area
4 cm
10 cm
Solution
a)
10 cm
4 cm
• the number of square
units needed to cover
a 3-D object
• the sum of the areas of
all the faces of an
object
6 cm
b)
The right rectangular prism has faces that are three different sizes.
front or back
4 cm
4 cm
6 cm
ends
top or bottom
10 cm
6 cm
10 cm
A=l×w
A = 10 × 6
A = 60
The area of the front
or back is 60 cm2.
c)
A=l×w
A = 10 × 4
A = 40
The area of the top
or bottom is 40 cm2.
A=l×w
A=6×4
A = 24
The area of each
end is 24 cm2.
Area is measured in
square units.
For example, square
centimetres, square
metres, etc.
The total surface area is the sum of the areas of all the faces.
The front and back
The top and bottom
The two ends have
have the same area:
have the same area:
the same area:
A = 60 × 2
A = 40 × 2
A = 24 × 2
A = 120
A = 80
A = 48
Total surface area = (area of front and back) + (area of top and bottom)
+ (area of ends)
You could add the areas you calculated
= 120 + 80 + 48
first. 60 + 40 + 24 = 124
= 248
Each area is the same as the area of one other
face, so you could then multiply the total by
The surface area of the right rectangular prism
two. 124 × 2 = 248
2
is 248 cm .
5.3 Surface Area of a Prism • MHR
19
What is the surface area of this right
rectangular prism?
16 cm
8 cm
3 cm
Example 2: Calculate the Surface Area of a Right Triangular
Prism
Draw the net of this right triangular prism.
b) What is the area of each face?
c) What is the total surface area?
a)
3m
a)
9m
Strategies
Draw a Diagram
Refer to page xxx.
Literacy
Link
An equilateral
triangle has three
equal sides and three
equal angles. Equal
sides are shown on
diagrams by placing
tick marks on them.
3m
2.6 m
b)
The bases of the prism are equilateral triangles.
The sides of the prism are rectangles.
rectangle
MHR • Chapter 5
triangle
2.6 m
3m
9m
A=l×w
A=9×3
A = 27
The area of one
rectangle is 27 m2.
20
2.6 m
9m
Solution
3m
A = (b × h) ÷ 2
A = (3 × 2.6) ÷ 2
A = 7.8 ÷ 2
A = 3.9
The area of one
triangle is 3.9 m2.
c)
You must add the areas of all faces to find the surface area.
This right triangular prism has five faces.
There are three rectangles of the same size and two triangles
of the same size.
Total surface area = (3 × area of rectangle) + (2 × area of triangle)
= (3 × 27) + (2 × 3.9)
= 81 + 7.8
= 88.8
The surface area of the right triangular prism is 88.8 m2.
Find the surface area of this
triangular prism.
9.9 cm
7 cm
2 cm
7 cm
• Surface area is the sum of the areas of all the faces of a 3-D object.
A1
A6
A2
A5
A3
A4
Surface Area = A1 + A2 + A3 + A4 + A5 + A6,
where A1 represents the area of rectangle 1, A2
represents the area of rectangle 2, etc.
1.
Write a set of guidelines that you could use to find the surface area
of a prism. Share your guidelines with a classmate.
2.
A right rectangular prism has six faces. Why do you have to find
the area of only three of the faces to be able to find the surface
area. Use pictures and words to explain your thinking.
5.3 Surface Area of a Prism • MHR
21
7.
For help with #3 and #4, refer to Example 1 on
page xxx.
3.
4.
Find the surface
area of this right
rectangular prism to
the nearest tenth of a
square centimetre.
Given the area of each face of a right
rectangular prism, what is the surface area?
front
12
20
13.5 cm
5 cm
8.
Find the surface area
of this CD case.
Paco builds a glass greenhouse.
1.1 m
2.4 m
1.8 m
1 cm
Calculate the
surface area of
2.7 m
this ramp in the
1.5 m
shape of a right
triangular prism.
2.3 m
Give your answer
to the nearest tenth of a square metre.
How many glass faces does the
greenhouse have?
b) How much glass does Paco need to buy?
3 cm
9 cm
22
MHR • Chapter 5
9.
What is the minimum amount of material
needed to make the cover of this textbook
if there is no overlap? Give your answer to
the nearest square millimetre.
10.
Jay wants to make a bike ramp. He draws
the following sketch. What is the surface
area of the ramp?
0.7 m
Cheese is sometimes packaged in a
triangular box. How much cardboard
would you need to cover this piece of
cheese if you do not include overlapping?
Calculate your answer to the nearest tenth
of a square centimetre.
6.4 cm
0.6 m
a)
For help with #5 to #7, refer to Example 2 on
page xxx.
4.5 cm
15 mm2
18.5 cm
12.3 cm
6.
side
mm2
mm2
14 cm
5.
top
The tick marks on
the two sides of
the triangle
indicate that these
sides are equal.
0.9 m
2.2 m
2m
1.6 m
11.
Dallas wants to paint three cubes. The
cubes measure 1 m × 1 m × 1 m,
2 m × 2 m × 2 m, and 3 m × 3 m × 3 m,
respectively. What total surface area will
Dallas paint if he decides not to paint the
bottoms of the three cubes?
14.
12.
Tadika has a gift to wrap. Both of these
containers will hold her gift. Which
container would allow her to use the
least amount of wrapping paper? Explain
your choice.
15. a)
If the edge length of a cube is doubled,
find the ratio of the old surface area to
the new surface area.
b) What happens if the edge length of
a cube is tripled? Is there a pattern?
7 cm
16.
30 cm
5 cm
10 cm
13.
10 cm
5 cm
A square cake pan measures 30 cm on
each side and is 5 cm deep. Cody wants to
coat the inside of the pan with non-stick
oil. If a single can of non-stick oil covers
an area of 400 000 cm2, how many pans
can be coated with a single can?
Ethan is hosting games night this weekend.
He bought ten packages of playing cards.
Each package measures 9 cm × 6.5 cm ×
1.7 cm. He wants to build a container to
hold all ten packages of cards.
a) What are the minimum inside
dimensions of the container?
b) Is there more than one kind of
container that would work? Draw
diagrams to help explain your answer.
Shelby wants
to paint the
walls and
ceiling of a
rectangular
room.
Type of Paint
2.6 m
6.8 m
4.8 m
Size of Paint Can
Cost
Wall paint
4L
1L
$24.95
$7.99
Ceiling paint
4L
$32.95
One litre of paint covers 9.5 m2.
a) What is the least amount of paint Shelby
can buy to paint the room (subtract 5 m2
for the door and windows)?
b) How much will the paint cost,
including the amount of tax charged
in your region?
MATH LINK
For the prism-shaped building you created in the Math Link on page XXX, how much
material do you need to cover the exterior walls and the roof of the building?
5.3 Surface Area of a Prism • MHR
23
Surface Area of a Cylinder
Focus on…
After this lesson,
you will be able to...
φ find the surface
area of a cylinder
Glow sticks work because of a chemical reaction. There are two
solutions in separate compartments inside the stick. Once you bend the
stick, the two solutions mix. This mixture creates a new solution that
gives off light. The colour of the glow stick depends on the dye in the
mixture. How might you determine how much plastic would be needed
to make a glow stick to fit around your wrist?
cylinder
• a three-dimensional
object with two
parallel and congruent
circular bases
How do you find the surface area of a right cylinder ?
1.
2.
Draw the net of a glow stick. Use the
actual dimensions from the diagram
shown.
List the shapes that make up your net.
3. a)
Copy and complete the table.
Area Formula
cylinder
Circle: A = × Rectangle: A = × 24
MHR • Chapter 5
21 cm
Use 3.14 as an
approximate value for π.
Measurements
π = 3.14
r=
l=
w=
The radius of a circle is
one half the diameter.
What measurement are you missing to calculate the area of each
shape? Use your piece of paper as a model of a glow stick to help
you visualize what might be missing.
c) How can you use what you know about circles to help you find
the missing measurement?
b)
If your calculator has
a π key, you can use
it to get a more
accurate answer.
d = 0.5 cm
The cylinder has two identical circles, one at each end.
What is the area of each circle?
b) What is the area of the rectangle?
c) The total surface area is the sum of the areas of all the shapes.
What is the surface area of the glow stick? Include the units in
your final answer.
4. a)
Pop cans are
cylinders. The world’s
largest Coke™ can is
located in Portage la
Prairie, Manitoba.
Reflect on Your Findings
How would you find the surface area of any right cylinder?
b) What type of units do you use to measure surface area?
5. a)
Example 1: Determine the Surface Area of a Right Cylinder
Estimate the surface area of the can.
b) What is the surface area of the can? Express
your answer to the nearest hundredth of a
square centimetre?
a)
11 cm
Solution
7.5 cm
The surface area of the can is found by adding the areas of the two
circular bases and the rectangular side that surrounds them.
The width, w, of the rectangle is the height of the can.
The length, l, of the rectangle is equal to the circumference of the circle.
a) To estimate, use approximate values: d ≈ 8 cm, w ≈ 10 cm, π ≈ 3.
Area of circle = π × r2
r2 means
≈3×4×4
r×r
The radius of a circle
≈ 48
is half the diameter.
There are two circles: 2 × 48 = 96
The area of the two circles is approximately 96 cm2.
Area of rectangle = l × w
Replace l with the formula for the
= (π × d) × w
≈ 3 × 8 × 10
circumference of a circle.
≈ 240
The area of the rectangle is approximately 240 cm2.
Estimated surface area = area of two circles + area of rectangle
≈ 96 + 240
≈ 340
The estimated surface area is 340 cm2.
Literacy
Link
circle
radius
centre
diameter
The formula for the
circumference of a circle is
C = π × d or C = 2 × π × r.
5.4 Surface Area of a Cylinder • MHR
25
Strategies
Draw a Diagram
Refer to page xxx.
b)
Method 1: Use a Net
Draw the net and label the measurements.
top
side
bottom
7.5 cm
11 cm
The diameter of the circle is 7.5 cm.
Determine the radius.
7.5 ÷ 2 = 3.75
The radius of the circle is 3.75 cm.
Find the area of one circle.
Use 3.14 as an
A = π × r2
approximate value
A ≈ 3.14 × 3.752
for π.
A ≈ 44.15625
The area of one circle is approximately 44.15625 cm2.
Find the area of two circles.
2 ×44.15625 = 88.3125
The area of both circles is approximately 88.3125 cm2.
Find the area of the rectangle using the circumference of the circle.
A=l×w
A = (π × d) × w
Replace l with the formula for the circumference of a circle.
A ≈ 3.14 × 7.5 × 11
A ≈ 259.05
The area of the rectangle is approximately 259.05 cm2.
Round your
answer at the end of
the calculation.
26
MHR • Chapter 5
Calculate the total surface area.
Total surface area = 88.3125 + 259.05
= 347.3625
The total surface area is approximately 347.36 cm2.
Method 2: Use a Formula.
Use this formula to find the total surface area of any cylinder.
S.A. = 2 × (π × r2) + (π × d × h)
This formula incorporates each shape and its area
S.A. ≈ 2 × (3.14 × 3.752) + (3.14 × 7.5 × 11)
formula to find the surface area.
2
× (π × r2) +
(π × d) × h
S.A. ≈ 88.3125 + 259.05
two
circles
circle
area
rectangle
area
S.A. ≈ 347.3625
formula
formula (length is the
2
The total surface area is 347.36 cm , to the
circumference of a circle;
nearest hundredth.
width is the height of
the cylinder)
Calculate the surface area of this
cylinder to the nearest tenth of a
square centimetre.
Literacy
Link
The abbreviation S.A.
is often used as a
short form for surface
area.
9 cm
55 cm
Example 2: Use the Surface Area of a Cylinder
Calculate the surface area of this totem pole, including the two circular
bases. The pole stands 2.4 m tall and has a diameter of 0.75 m. Give
your answer to the nearest hundredth of a square metre.
Solution
The cylinder has two circular bases.
The area of one circle is:
A = π × r2
r=d÷ 2
A ≈ 3.14 × 0.3752
A ≈ 0.4415625
The area of the circle is
approximately 0.4415625 m2.
There are two circles, so the area
of both circles is approximately
0.883125 m2.
The side of the cylinder is
a rectangle.
The area of the rectangle is:
A = (π × d) × h
A ≈ 3.14 × 0.75 × 2.4
A ≈ 5.652
The area of the rectangle is
approximately 5.652 m2.
Calculate the total surface area.
S.A. ≈ 0.883125 + 5.652
S.A. ≈ 6.535125
The total surface area is approximately 6.54 m2.
Replace one dimension
with the formula for the
circumference of a circle.
Calculate the surface area of a cylindrical waste backet without a lid
that measures 28 cm high and 18 cm in diameter. Give your answer
to the nearest square centimetre.
This metal totem pole
was created by Todd
Baker, Squamish Nation.
It represents the Birth of
the Bear Clan, with the
princess of the clan on
the top half and the
bear on the bottom half.
5.4 Surface Area of a Cylinder • MHR
27
• The surface area of a cylinder is the sum of the
areas of its faces.
• A net of a cylinder is made up of one
rectangle and two circles.
• To find one of the dimensions of the rectangle,
calculate the circumference of the circle.
The length of this side
is the circumference of
the circle C = π ⴛ d or
C=2ⴛπⴛr
1.
What are the similarities and differences between finding the
surface area of a prism and finding the surface area of a cylinder?
2.
Explain why you need to find the circumference of a circle to find
the surface area of a cylinder.
5.
For help with #3 to #7, refer to Examples 1 and 2 on
pages xxx–xxx.
Find the surface area of each object
to the nearest tenth of a square unit.
a)
d = 2.5 cm
b)
d = 0.003 m
16 cm
Draw a net for this cylinder.
b) Sketch a different net for
this cylinder.
3. a)
4.
wooden rod
16 m
Estimate the surface area of each cylinder.
Then, calculate each surface area to the
nearest tenth of a square centimetre.
a)
b)
d = 7 cm
30 cm
r = 10 cm
22 cm
flag pole
6.
Use the formula
S.A. = 2 × (π × r2) + (π × d × h) to
calculate the surface area of each object.
Give each answer to the nearest hundredth
of a square unit.
a) d = 2.5 cm
b)
d = 5 cm
10 cm
You can simplify the formula:
S.A. =2 × (π × r2) + (π × d × h)
= 2πr2 + πdh
28
MHR • Chapter 5
7 cm
7.
8.
Do you prefer to find the surface area of
a cylinder by using the sum of the area of
each face or by using a formula? Give at
least two reasons for your choice.
11.
If each tennis ball has a diameter of 7 cm,
calculate the amount of material needed to
make a can that holds three tennis balls.
12.
Coins can be stored in a plastic wrapper
similar to a cylinder. A roll of dimes
contains 50 coins. Each dime has a
diameter of 17.5 mm and a thickness of
1 mm. Calculate the minimum surface
area of the plastic wrapper.
13.
A paint roller in the shape of a cylinder
with a radius of 4 cm and a length of
21 cm is rolled vertically on a wall.
a) What is the length and width of the
wet path after ten complete rolls?
b) What area does the paint cover?
Anu wants to re-cover the cylindrical stool
in his bedroom. How much material does
he need if there is no overlap?
d = 42 cm
32 cm
9.
Kaitlyn and Hakim each bought a tube
of candy. Both containers cost the same
amount. Which container required more
plastic to make?
d = 7 cm
CANDY
122 cm
d = 11 cm
CANDY
85 cm
10.
Paper towel is rolled
around a cardboard
tube. Calculate the
outside surface area
of the tube.
r = 2 cm
Each person
produces about
1.59 kg of trash each
day. Most of this is
paper products.
27.5 cm
MATH LINK
For the cylindrical building you created in the Math
Link on page XXX, how much material do you need to
cover the exterior walls and the roof of the building?
Douglas J. Cardinal, one of the world’s most
acclaimed architects, uses his European, Blackfoot,
and Ojibwa roots when designing buildings. He is
known for his design of The Canadian Museum of
Civilization in Gatineau, Québec, as well as a
number of buildings in Western Canada, such as
Telus World of Science in Edmonton and First
Nations University of Canada in Regina.
5.4 Surface Area of a Cylinder • MHR
29
Key Words
8.
Unscramble the letters for each puzzle in #1 to #6.
Use the clues to help you solve the puzzles.
1.
E T N
a flat diagram that you can fold to make
a 3-D object
2.
U S F A R E C E R A A
the sum of the areas of the faces of an
object (2 words)
3.
I R H T G R P M S I
a prism whose sides are perpendicular
to its bases (2 words)
4.
E C N I Y D R L
a 3-D object with two parallel circular
bases
5.
I R A G N R U A L T S I M R P
a 3-D object with two parallel triangular
bases (2 words)
6.
L E U C A A N R G T R I R M S P
a 3-D object with two parallel rectangular
bases (2 words)
5.1 Views of Three-Dimensional Objects,
pages xxx–xxx
7. Draw and label the top, front, and side
views for these objects.
a)
30
b)
MHR • Chapter 5
9.
Using isometric paper, draw each
3-D object from the views given.
a)
front
top
side
b)
front
top
side
A filing cabinet is in the far corner of an
office. Shay is redecorating the room and
wants to turn the cabinet 90° clockwise.
Here are the views before the turn:
front
top
side
How does each view change after
the turn?
b) Draw and label the top, front, and
side views of the filing cabinet after
it is turned.
a)
5.2 Nets of Three-Dimensional Objects,
pages xxx–xxx
10. Name the object formed by each net.
a)
b)
c)
15.
Find the surface area of each triangular
prism.
a)
10 cm
6 cm
4 cm
8 cm
11.
Draw the net for each object.
a)
b)
b)
22.9 cm
50 cm
COOKIES
SOUP
20 cm
22.5 cm
12.
Using two pieces of grid paper, create a
pencil box and lid. Draw a net, cut it out,
fold it, and build your pencil box. Make
sure new pencils fit in it!
16.
Liza had two more gifts left to wrap when
she ran out of paper. Approximately how
much more wrapping paper does she need
to finish wrapping her gifts? Assume no
overlap.
5.3 Surface Area of a Prism, pages xxx–xxx
20.5 cm
For #13 to #16, calculate the surface area to the
nearest tenth of a square unit.
13.
32.5 cm
23 cm
12.5 cm
a)
35 cm
12 cm
12 cm
12 cm
5.4 Surface Area of a Cylinder,
pages xxx–xxx
For #17 to #19, calculate the surface area to
the nearest tenth of a square unit.
b)
1.7 m
0.5 m
2m
14.
12.5 cm
What is the surface area of each object?
17.
Determine the surface
area of the cylinder.
2.5 m
5.5 m
Using the measurements shown on the net
of the rectangular prism, calculate the
surface area.
10 mm
27 mm
42 mm
18.
The pencil sharpener on Kay’s desk has
a diameter of 3.4 cm and is 7 cm tall.
Calculate the surface area.
19.
The circumference of a container’s lid is
157 cm. If the container is 102 cm tall,
what is the surface area of the container?
Chapter Review • MHR
31
For #1 to #5, choose the best answer.
Short Answer
6. Sketch the top, front, and side views
of this building.
The top view of this
container shows a
A circle
B square
C triangle
D rectangle
2.
One face on a cube has an area of 49 cm2.
What is the surface area of the cube?
A 343 cm2
B 294 cm2
C 196 cm2
D 154 cm2
3.
What three-dimensional
object has a net like
this one?
A cube
B cylinder
C triangular prism
D rectangular prism
4.
What is the surface
area of this box?
A 550 mm2
B 900 mm2
C 1100 mm2
D 1800 mm2
CRACKERS
1.
5 mm
20 mm
18 mm
7.
An object may have more than one net.
Draw three different nets for this cube.
8.
A DVD case is made of a plastic covering
that measures 19 cm long, 13.5 cm wide,
and 1.4 cm thick. Calculate the surface
area to the nearest tenth of a square
centimetre.
19 cm
5.
32
What is the surface area of a cylinder that
is 30.5 cm long and has a radius of 3 cm,
to the nearest hundredth of a square
centimetre?
A 274.50 cm2
B 603.19 cm2
C 631.14 cm2
D 688.01 cm2
MHR • Chapter 5
13.5 cm
1.4 cm
9.
The surface area of a cube is 1014 cm2.
Find the length of any side of the cube.
Extended Response
10. a) Sketch a three-dimensional object you
can build using two of these triangular
prisms.
12.
Single-serving juice boxes measure 10 cm
by 7 cm by 5 cm. A manufacturer wants
to shrink wrap four boxes together for
sale. Which of the following arrangements
of the boxes will use the least amount of
plastic wrap? Show how you know.
10 cm
5 cm
7 cm
Arrangement 1
Draw the front view, top view, and
side view of your object.
c) Draw a net for your object.
b)
11.
Ken and Arika are comparing their
cylinders. Arika’s cylinder is twice as tall
as Ken’s, but is only half the diameter.
Ken’s cylinder has a height of 18 cm and a
diameter of 9 cm. Whose cylinder has the
greater surface area?
10 cm
5 cm
7 cm
Arrangement 2
WRAP IT UP!
It is time to create your miniature community!
Work together to finalize one aerial view for
your community. You may choose to start
with one that you created on page XXX.
Include the following in your diagram and
description:
• All the buildings designed by you and your
group members.
• A 3-D sketch, net, and surface area
calculations for one new building for each
member of your group. The new designs
should include at least one prism and cylinder.
Check each other’s work before submitting.
• Streets to navigate through the city.
• Environmental considerations such as water
source, parks, etc.
Practice Test • MHR
33
Let’s Face It!
1.
2.
34
Play Let’s Face It! with a partner or in a small group. These are
the rules:
• Remove the jacks, queens, kings, aces, and jokers from the deck
of cards.
• Take turns dealing the cards. It does not matter who deals first.
• Shuffle the cards and deal three cards, face up, to each player.
• Use the values of the cards as the dimensions, in centimetres,
of a rectangular prism.
• Calculate the surface area of your rectangular prism using
pencil and paper.
• Each player who calculates the surface area of their prism
correctly scores a point. (You will need to check each other’s
work.)
• The player with the rectangular prism that has the greatest
surface area scores an extra point for that round. If there is
a tie, each of the tied players scores an extra point.
• The first player to reach ten points wins the game. If more than
one player earns ten points in the same game, these players
continue playing until one of them pulls ahead.
Play a different version of Let’s Face It! by modifying the rules
as follows:
• Deal only two cards to each player and use them to describe the
size of a right cylinder. The first card gives the radius of each
circle, in centimetres. The second card gives the height of the
cylinder, in centimetres.
• Use a calculator to determine the surface area of your cylinder,
to the nearest hundredth of a square centimetre.
• Award points and decide the winner in the same way as before.
MHR • Chapter 5
• deck of playing cards
per pair or small
group
• calculator per student
My cards are a 5 of clubs,
a 3 of hearts, and an 8 of
spades. My rectangular
prism has edges of 5 cm,
3 cm, and 8 cm.
3 cm
8 cm
5 cm
I was dealt a 4 of clubs
and then a 6 of clubs.
The radius of each circle
is 4 cm. The height of
the cylinder is 6 cm.
4 cm
6 cm
Design a Bedroom
Have you ever wondered what it
would be like to completely
design a room? Suppose you were
given the opportunity to create
the kind of space that a person
your age would appreciate and
make good use of.
You be the interior designer. Your
first project is to create a design
for a bedroom that is 4 m wide
by 5 m long, and is 2.5 m high.
Draw the top view of the
room and place at least
three objects in the room.
b) Draw the top, front, and
side views of at least three
objects you put in the
room. Identify the 3-D
shape that each object
closely resembles.
1. a)
Painting your room is the
next step. Determine the
amount of paint you need
to cover the walls and
ceiling of your room.
b) One can of the paint you
are going to use covers
10 m2/L. How many cans
do you need?
2. a)
Challenge in Real Life • MHR
35